p dirichlet

Dirichlet's approximation theorem

In mathematics, Dirichlet's theorem on diophantine approximation, also called Dirichlet's approximation theorem, states that for any real number α, and any positive integer n, there is some positive integer mn , such that the difference between mα and the nearest integer is at most . This is a consequence of the pigeonhole principle.

For example, no matter what value is chosen for α, at least one of the first five integer multiples of α, namely

1α, 2α, 3α, 4α, 5α,

will be within of an integer, either above or below. Likewise, at least one of the first 20 integer multiples of α will be within of an integer.

Dirichlet's approximation theorem shows that the Thue–Siegel–Roth theorem is the best possible in the sense that the occurring exponent cannot be increased, and thereby improved, to −2.

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